Following the notations in [1]}, consider the following diagram
\({E_{\mathbf {V}_{\alpha }}\times E_{\mathbf {V}_{\beta }} & E_{\alpha ,\beta }^{\prime }[r]^-{p_{2\alpha ,\beta }}[l]_-{p_{1\alpha ,\beta }} & E_{\alpha ,\beta }^{\prime \prime }[r]^-{p_{3\alpha ,\beta }} & E_{\mathbf {V}_{\alpha +\beta }}\\& & \tilde{F}[u]_-{\tilde{\iota }}[r]^-{\tilde{p}_3} & F_{mi,\alpha +\beta -mi}[u]_-{\iota _{mi,\alpha +\beta -mi}}[d]^-{\kappa _{mi,\alpha +\beta -mi}}\\& & & E_{\mathbf {V}_{\beta ^{\prime }},}}\) 
where \(\tilde{F}=E_{\alpha ,\beta }^{\prime \prime }\times _{E_{\mathbf {V}_{\alpha +\beta }}}F_{mi,\alpha +\beta -mi}\) 
and the following diagram is a Cartesian diagram
\({E_{\alpha ,\beta }^{\prime \prime }[r]^-{p_{3\alpha ,\beta }} & E_{\mathbf {V}_{\alpha +\beta }}\\\tilde{F}[u]_-{\tilde{\iota }}[r]^-{\tilde{p}_3} & F_{mi,\alpha +\beta -mi}[u]_-{\iota _{mi,\alpha +\beta -mi}}.}\) 
By definition, \(\tilde{F}=\lbrace (x,\mathbf {W})\rbrace \) , where \(x\in E_{{\mathbf {V}}_{\alpha +\beta }}\)  and \(\mathbf {W}\subset {\mathbf {V}}_{\alpha +\beta }\)  is a subspace with dimension vector \(\beta \)  such that \(\mathbf {W}\)  and \(\mathbf {W}_{\beta ^{\prime }}\)  are \(x\) -stable ([1]}).
Now,
\({_{mi}\mathcal {R}}(A\ast B)&=&(\kappa _{mi,\alpha +\beta -mi})_!\iota _{mi,\alpha +\beta -mi}^\ast (p_{3\alpha ,\beta })_!(p_{2\alpha ,\beta })_\flat p_{1\alpha ,\beta }^{\ast }(A\boxtimes B)[M](\frac{M}{2})\\&=&(\kappa _{mi,\alpha +\beta -mi})_! (\tilde{p}_{3})_!\tilde{\iota }^\ast (p_{2\alpha ,\beta })_\flat p_{1\alpha ,\beta }^{\ast }(A\boxtimes B)[M](\frac{M}{2}),\) 
where \(M=m_{\alpha ,\beta }-\langle {mi,\alpha +\beta -mi}\rangle \) .
Let
\(\tilde{F}_{\lambda _t}=\lbrace (x,\mathbf {W})\in \tilde{F}\,\,|\,\,\underline{\mbox{\rm dim}\,}(\mathbf {W}\cap \mathbf {W}_{\beta ^{\prime }})=\beta -(m-t)i\rbrace .\) 
It is clear that \(\tilde{F}\)  is the disjoint of \(\tilde{F}_{\lambda _t}\) .
Let \(\tilde{F}_{\leqslant t}=\bigcup _{l=a}^{t}\tilde{F}_{\lambda _l}\)  and \(\tilde{F}_{\geqslant t}=\bigcup _{l=t}^{b}\tilde{F}_{\lambda _l}\) . Then \(\tilde{F}_{\leqslant t}\)  is open
and \(\tilde{F}_{\geqslant t}\)  is closed in \(\tilde{F}\) . At the same time, \(\tilde{F}_{\lambda _t}\)  is closed in \(\tilde{F}_{\leqslant t}\)  and open in \(\tilde{F}_{\geqslant t}\) , respectively.
These embeddings are denoted by \(j_{\lambda _t}:\tilde{F}_{\lambda _t}\rightarrow \tilde{F}\) , \(j_{\leqslant t}:\tilde{F}_{\leqslant t}\rightarrow \tilde{F}\)  and \(j_{\geqslant t}:\tilde{F}_{\geqslant t}\rightarrow \tilde{F}\) .
For any complex \(L\)  on \(\tilde{F}\) , we have the following distinguished triangles (Section 1.4.5 in [3]})
\((j_{{\lambda _t}})_!j^\ast _{\lambda _t}L\rightarrow (j_{\geqslant t})_!j^\ast _{_{\geqslant t}}L\rightarrow (j_{\geqslant t+1})_!j^\ast _{_{\geqslant t+1}}L\rightarrow .\) 
When \(L=\tilde{\iota }^\ast (p_{2\alpha ,\beta })_\flat p_{1\alpha ,\beta }^{\ast }(A\boxtimes B)\) ,
we get the following distinguished triangles
\({_{mi}\mathcal {R}}(A\ast B)_{\lambda _t}\rightarrow {_{mi}\mathcal {R}}(A\ast B)_{\geqslant t}\rightarrow {_{mi}\mathcal {R}}(A\ast B)_{\geqslant t+1}\rightarrow ,\) 
where
\({_{mi}\mathcal {R}}(A\ast B)_{\lambda _t}=(\kappa _{mi,\alpha +\beta -mi})_! (\tilde{p}_{3})_!(j_{{\lambda _t}})_!j^\ast _{\lambda _t}\tilde{\iota }^\ast (p_{2\alpha ,\beta })_\flat p_{1\alpha ,\beta }^{\ast }(A\boxtimes B)[M](\frac{M}{2})\) 
and
\({_{mi}\mathcal {R}}(A\ast B)_{\geqslant t}=(\kappa _{mi,\alpha +\beta -mi})_! (\tilde{p}_{3})_!(j_{\geqslant t})_!j^\ast _{\geqslant t}\tilde{\iota }^\ast (p_{2\alpha ,\beta })_\flat p_{1\alpha ,\beta }^{\ast }(A\boxtimes B)[M](\frac{M}{2}).\) 
In the proof of Theorem 3.1 in [1]}, Fang-Lan-Xiao proved that
\(&&{_{mi}\mathcal {R}}(A\ast B)_{\lambda _t}\\&=&(\kappa _{mi,\alpha +\beta -mi})_! (\tilde{p}_{3})_!(j_{{\lambda _t}})_!j^\ast _{\lambda _t}\tilde{\iota }^\ast (p_{2\alpha ,\beta })_\flat p_{1\alpha ,\beta }^{\ast }(A\boxtimes B)[M](\frac{M}{2})\nonumber \\&\simeq &(L_{ti}\ast L_{(m-t)i})\boxtimes ({_{ti}\mathcal {R}}(A)\ast {_{(m-t)i}\mathcal {R}}(B))[-P_t](-\frac{P_t}{2})\nonumber \\&\simeq &L^{f_{m,t}(v)}_{mi}\boxtimes ({_{ti}\mathcal {R}}(A)\ast {_{(m-t)i}\mathcal {R}}(B))[-P_t](-\frac{P_t}{2})\nonumber \\&\simeq &\mathbf {1}_{pt}^{f_{m,t}(v)}\boxtimes ({_{ti}\mathcal {R}}(A)\ast {_{(m-t)i}\mathcal {R}}(B))[-P_t](-\frac{P_t}{2}).\nonumber \) 
Formulas (REF ) and (REF ) imply the desired result.
